Manifolds of nonpositive curvature.

*(English)*Zbl 0591.53001
Progress in Mathematics, 61. Boston-Basel-Stuttgart: Birkhäuser. iv, 263 pp. SFr. 74.00; DM 88.00 (1985).

This book is based on 4 lectures given by M. Gromov at the Collège de France, Paris, in February of 1981. The text was written by V. Schroeder after further consultations with Gromov. The book also includes five appendices, including four by Schroeder and one by W. Ballmann, that describe results related to the main themes of the book.

The inspiration for the two main results described below comes primarily from the work of D. Kazhdan and G. Margulis [Mat. Sb., Nov. Ser. 75(117), 163–168 (1968; Zbl 0241.22024)], A. Borel [Sous-groupes discréts de groupes semisimples, Sém. Bourbaki 1968/69, Exposé No. 358, 199–215 (1971; Zbl 0225.22017)] and the Strong Rigidity Theorem of G. D. Mostow [Strong rigidity of locally symmetric spaces (Ann. Math. Studies 78) (1973; Zbl 0265.53039)]. The results of these authors that are relevant here may be stated in terms of the geometry of symmetric spaces of noncompact type, and the problem considered by Gromov is to what extent these results can be generalized to arbitrary manifolds of nonpositive sectional curvature. Perhaps surprisingly it turns out that the real analyticity of a symmetric space of noncompact type is sometimes just as important for such extensions as the nonpositivity of the sectional curvature; see Theorem 1. We now state the main results:

Theorem 1. For each positive integer \(n\geq 2\) there exist positive constants \(\delta(n)\) and \(\varepsilon(n)\) with the following property: Let M be an n-dimensional real analytic complete Riemannian manifold whose sectional curvature satisfies \(-1\leq K\leq 0\) and whose injectivity radius tends to zero as a point in \(M\) tends to infinity. Then

(1) \(M\) is diffeomorphic to the interior of a compact manifold with boundary.

(2) The volume of \(M\) is at least \(\delta(n)\).

(3) If \(M\) has no Euclidean local de Rham factor and if \(\beta_ i\) denotes the \(i\)-th Betti number of \(M\), then \(\sum \beta_ i\leq [\varepsilon(n)-\text{vol}(M)]\) where \(\varepsilon(n)-\text{vol}(M)\) denotes the largest number of disjoint imbedded geodesic \(\varepsilon(n)\)-balls in \(M\).

Theorem 2. Let \(M^*\) be a compact irreducible locally symmetric space of nonpositive sectional curvature whose \({\mathbb R}\)-rank is at least 2. Let \(M\) be a compact \(C^{\infty}\) manifold with nonpositive sectional curvature whose fundamental group is isomorphic to that of \(M^*\). Then \(M\) is isometric to \(M^*\) after rescaling the metric on \(M^*\) by suitable constants on each local de Rham factor.

Theorem 1 applies to manifolds of finite volume. Assertion 2 in Theorem 1 for locally symmetric spaces of noncompact type and finite volume is a corollary of a result of Kazhdan and Margulis and Borel (loc. cit.). The idea of the proof of assertion 1 of Theorem 1 has previously been used to prove the same result for \({\mathbb{R}}\)-rank 1 locally symmetric spaces of finite volume - see for example H. Garland and M. Raghunathan [Ann. Math. (2) 92, 279–326 (1970; Zbl 0206.03603)], M. Raghunathan [Discrete subgroups of Lie groups (1972; Zbl 0254.22005), or Invent. Math. 4, 318–338 (1968; Zbl 0218.22015)]. Moreover, one can give a more precise description of the structure of \(M\) outside a large compact set if \(M\) has finite volume and is either locally symmetric of any \({\mathbb R}\)-rank or has strictly negative sectional curvature [A. Borel, Introduction aux groupes arithmétiques (1969; Zbl 0186.33202), the reviewer, Ann. Math. (2) 111, 435–476 (1980; Zbl 0401.53015), H. Garland and M. Raghunathan, Ann. Math. (2) 92, 279–326 (1970; Zbl 0206.03603), E. Heintze, Mannigfaltigkeiten negativer Krümmung, Habilitationsschrift, Univ. Bonn (1976), G. Prasad, Invent. Math. 21, 255–286 (1973; Zbl 0264.22009)]. Assertion 3 of Theorem 1 is new in this context, but Gromov has proven an analogous result for compact manifolds of nonnegative sectional curvature – see (0.2.A) of M. Gromov [Comment. Math. Helv. 56, 179–195 (1981; Zbl 0467.53021)].

The idea of the proof of assertion 1 of Theorem 1 is to construct a Morse function on \(M\) that has no critical points outside a compact subset of \(M\). A closer analysis of this function will then establish assertions 2 and 3. A first approximation of the desired Morse function is the function \(\text{Inj Rad}: p\to \text{injectivity radius of }M \text{ at }p\). Although not smooth the function \(\text{Inj Rad}\) is proper and if the sectional curvature is negative then \(\text{Inj Rad}\) has no critical points in a certain generalized sense outside a compact subset of \(M\). Assertions 1 and 2 of Theorem 1 follow from considering \(\text{Inj Rad}\) alone in the case that \(K<0\) and \(M\) is merely \(C^{\infty}\) – see M. Gromov [Comment. Math. Helv. (loc. cit.)]. In the general case one constructs a smooth Morse function \(f: M\to [0,\infty)\) that is closely related to \(\text{Inj Rad}\). Theorem 1 is false in the \(C^{\infty}\) case with \(K\leq 0\) unless one excludes the existence of small totally geodesic flat 2-tori – see Appendix 2.

Theorem 2 is of course the Mostow rigidity theorem for \({\mathbb R}\)-rank\(\leq 2\) if both \(M^*\) and \(M\) are locally symmetric. The proof of Theorem 2 is accomplished by adapting essential parts of Mostow’s argument to the case at hand. Note that Theorem 2 implies that a compact, irreducible, locally symmetric space with \({\mathbb R}\)-rank \(\geq 2\) and nonpositive sectional curvature admits no other metrics of nonpositive sectional curvature except those obtained by multiplication of the given metric by positive constants on each local de Rham factor of \(M^*\). Other proofs of Theorem 2 exist now – see for example the reviewer [Ergodic Theory Dyn. Syst. 3, 47–85 (1983; Zbl 0521.53045), or Symmetry diffeomorphism group of a manifold of nonpositive curvature, Trans. Am. Math. Soc. 309, No. 1, 355–374 (1988; Zbl 0662.53035) and Indiana Univ. Math. J. 37, No. 4, 735–752 (1988; Zbl 0676.53054)], W. Ballmann [Ann. Math., II. Ser. 122, 597–609 (1985; Zbl 0585.53031)], K. Burns and R. Spatzier [Manifolds of nonpositive curvature and their buildings (preprint, 1985)].

Besides the two results stated above the book contains other noteworthy features, in particular the definition and study of the Tits metric on the boundary sphere \(\tilde M(\infty)\) and local rigidity phenomena. The Tits metric is a metric space structure defined on \(\tilde M(\infty)\) for any complete, simply connected manifold \(\tilde M\) of nonpositive sectional curvature. Two distinct points \(x,y\) of \(\tilde M(\infty)\) may have infinite Tits distance; for example, this is the case for any two distinct points \(x,y\) of \(\tilde M(\infty)\) whenever the sectional curvature of \(\tilde M\) is bounded above by a negative constant. However, for spaces with significant amounts of zero curvature the Tits metric on \(\tilde M(\infty)\) provides valuable information about the geometry of the underlying manifold \(\tilde M\). For example, one can tell from the intrinsic Tits geometry of \(\tilde M(\infty)\) if the manifold \(\tilde M\) is a Riemannian product (Appendix 4). It seems likely that symmetric spaces of noncompact type and \({\mathbb R}\)-rank\(\geq 2\) can be characterized by appropriate intrinsic properties of the Tits metric. At present one knows that if \(f: \tilde M_ 1(\infty)\to \tilde M_ 2(\infty)\) is a homeomorphism of sphere topologies that preserves the Tits metrics, then \(\tilde M_ 2\) is a symmetric space of noncompact type and \({\mathbb R}\)-rank \(\geq 2\) whenever \(\tilde M_ 1\) is such a space (Appendix 4). The Tits metric is an important new tool in the study of manifolds of nonpositive sectional curvature, and it seems evident that further applications will be found in the future.

In this book Gromov also considers problems of local rigidity; for example, given a simply connected manifold \((\tilde M,g)\) of nonpositive sectional curvature and a compact set \(C\subseteq \tilde M\), is it possible to find a new metric \(g^*\) of nonpositive sectional curvature on \(\tilde M\) that agrees with g on \(\tilde M-C\) and is not isometric to \(g\)? Gromov shows that the answer is negative if \(\tilde M\) satisfies a certain geometric condition which in turn is satisfied by flat Euclidean spaces and by symmetric spaces of noncompact type and \({\mathbb R}\)-rank \(\geq 2\). The basic ideas have recently been extended in V. Schroeder and W. Ziller [Local rigidity of symmetric spaces (preprint, 1986), Trans. Am. Math. Soc. 320, No. 1, 145–160 (1990; Zbl 0724.53033)] to yield similar results for \({\mathbb R}\)-rank 1 symmetric spaces of noncompact type [see also L. Green and R. Gulliver, J. Differ. Geom. 22, 43–47 (1985; Zbl 0563.53036)].

Beyond the scope of this book Gromov has also initiated a study of nonsmooth manifolds of nonpositive curvature that include polyhedra and orbifolds. A manuscript is in preparation.

The inspiration for the two main results described below comes primarily from the work of D. Kazhdan and G. Margulis [Mat. Sb., Nov. Ser. 75(117), 163–168 (1968; Zbl 0241.22024)], A. Borel [Sous-groupes discréts de groupes semisimples, Sém. Bourbaki 1968/69, Exposé No. 358, 199–215 (1971; Zbl 0225.22017)] and the Strong Rigidity Theorem of G. D. Mostow [Strong rigidity of locally symmetric spaces (Ann. Math. Studies 78) (1973; Zbl 0265.53039)]. The results of these authors that are relevant here may be stated in terms of the geometry of symmetric spaces of noncompact type, and the problem considered by Gromov is to what extent these results can be generalized to arbitrary manifolds of nonpositive sectional curvature. Perhaps surprisingly it turns out that the real analyticity of a symmetric space of noncompact type is sometimes just as important for such extensions as the nonpositivity of the sectional curvature; see Theorem 1. We now state the main results:

Theorem 1. For each positive integer \(n\geq 2\) there exist positive constants \(\delta(n)\) and \(\varepsilon(n)\) with the following property: Let M be an n-dimensional real analytic complete Riemannian manifold whose sectional curvature satisfies \(-1\leq K\leq 0\) and whose injectivity radius tends to zero as a point in \(M\) tends to infinity. Then

(1) \(M\) is diffeomorphic to the interior of a compact manifold with boundary.

(2) The volume of \(M\) is at least \(\delta(n)\).

(3) If \(M\) has no Euclidean local de Rham factor and if \(\beta_ i\) denotes the \(i\)-th Betti number of \(M\), then \(\sum \beta_ i\leq [\varepsilon(n)-\text{vol}(M)]\) where \(\varepsilon(n)-\text{vol}(M)\) denotes the largest number of disjoint imbedded geodesic \(\varepsilon(n)\)-balls in \(M\).

Theorem 2. Let \(M^*\) be a compact irreducible locally symmetric space of nonpositive sectional curvature whose \({\mathbb R}\)-rank is at least 2. Let \(M\) be a compact \(C^{\infty}\) manifold with nonpositive sectional curvature whose fundamental group is isomorphic to that of \(M^*\). Then \(M\) is isometric to \(M^*\) after rescaling the metric on \(M^*\) by suitable constants on each local de Rham factor.

Theorem 1 applies to manifolds of finite volume. Assertion 2 in Theorem 1 for locally symmetric spaces of noncompact type and finite volume is a corollary of a result of Kazhdan and Margulis and Borel (loc. cit.). The idea of the proof of assertion 1 of Theorem 1 has previously been used to prove the same result for \({\mathbb{R}}\)-rank 1 locally symmetric spaces of finite volume - see for example H. Garland and M. Raghunathan [Ann. Math. (2) 92, 279–326 (1970; Zbl 0206.03603)], M. Raghunathan [Discrete subgroups of Lie groups (1972; Zbl 0254.22005), or Invent. Math. 4, 318–338 (1968; Zbl 0218.22015)]. Moreover, one can give a more precise description of the structure of \(M\) outside a large compact set if \(M\) has finite volume and is either locally symmetric of any \({\mathbb R}\)-rank or has strictly negative sectional curvature [A. Borel, Introduction aux groupes arithmétiques (1969; Zbl 0186.33202), the reviewer, Ann. Math. (2) 111, 435–476 (1980; Zbl 0401.53015), H. Garland and M. Raghunathan, Ann. Math. (2) 92, 279–326 (1970; Zbl 0206.03603), E. Heintze, Mannigfaltigkeiten negativer Krümmung, Habilitationsschrift, Univ. Bonn (1976), G. Prasad, Invent. Math. 21, 255–286 (1973; Zbl 0264.22009)]. Assertion 3 of Theorem 1 is new in this context, but Gromov has proven an analogous result for compact manifolds of nonnegative sectional curvature – see (0.2.A) of M. Gromov [Comment. Math. Helv. 56, 179–195 (1981; Zbl 0467.53021)].

The idea of the proof of assertion 1 of Theorem 1 is to construct a Morse function on \(M\) that has no critical points outside a compact subset of \(M\). A closer analysis of this function will then establish assertions 2 and 3. A first approximation of the desired Morse function is the function \(\text{Inj Rad}: p\to \text{injectivity radius of }M \text{ at }p\). Although not smooth the function \(\text{Inj Rad}\) is proper and if the sectional curvature is negative then \(\text{Inj Rad}\) has no critical points in a certain generalized sense outside a compact subset of \(M\). Assertions 1 and 2 of Theorem 1 follow from considering \(\text{Inj Rad}\) alone in the case that \(K<0\) and \(M\) is merely \(C^{\infty}\) – see M. Gromov [Comment. Math. Helv. (loc. cit.)]. In the general case one constructs a smooth Morse function \(f: M\to [0,\infty)\) that is closely related to \(\text{Inj Rad}\). Theorem 1 is false in the \(C^{\infty}\) case with \(K\leq 0\) unless one excludes the existence of small totally geodesic flat 2-tori – see Appendix 2.

Theorem 2 is of course the Mostow rigidity theorem for \({\mathbb R}\)-rank\(\leq 2\) if both \(M^*\) and \(M\) are locally symmetric. The proof of Theorem 2 is accomplished by adapting essential parts of Mostow’s argument to the case at hand. Note that Theorem 2 implies that a compact, irreducible, locally symmetric space with \({\mathbb R}\)-rank \(\geq 2\) and nonpositive sectional curvature admits no other metrics of nonpositive sectional curvature except those obtained by multiplication of the given metric by positive constants on each local de Rham factor of \(M^*\). Other proofs of Theorem 2 exist now – see for example the reviewer [Ergodic Theory Dyn. Syst. 3, 47–85 (1983; Zbl 0521.53045), or Symmetry diffeomorphism group of a manifold of nonpositive curvature, Trans. Am. Math. Soc. 309, No. 1, 355–374 (1988; Zbl 0662.53035) and Indiana Univ. Math. J. 37, No. 4, 735–752 (1988; Zbl 0676.53054)], W. Ballmann [Ann. Math., II. Ser. 122, 597–609 (1985; Zbl 0585.53031)], K. Burns and R. Spatzier [Manifolds of nonpositive curvature and their buildings (preprint, 1985)].

Besides the two results stated above the book contains other noteworthy features, in particular the definition and study of the Tits metric on the boundary sphere \(\tilde M(\infty)\) and local rigidity phenomena. The Tits metric is a metric space structure defined on \(\tilde M(\infty)\) for any complete, simply connected manifold \(\tilde M\) of nonpositive sectional curvature. Two distinct points \(x,y\) of \(\tilde M(\infty)\) may have infinite Tits distance; for example, this is the case for any two distinct points \(x,y\) of \(\tilde M(\infty)\) whenever the sectional curvature of \(\tilde M\) is bounded above by a negative constant. However, for spaces with significant amounts of zero curvature the Tits metric on \(\tilde M(\infty)\) provides valuable information about the geometry of the underlying manifold \(\tilde M\). For example, one can tell from the intrinsic Tits geometry of \(\tilde M(\infty)\) if the manifold \(\tilde M\) is a Riemannian product (Appendix 4). It seems likely that symmetric spaces of noncompact type and \({\mathbb R}\)-rank\(\geq 2\) can be characterized by appropriate intrinsic properties of the Tits metric. At present one knows that if \(f: \tilde M_ 1(\infty)\to \tilde M_ 2(\infty)\) is a homeomorphism of sphere topologies that preserves the Tits metrics, then \(\tilde M_ 2\) is a symmetric space of noncompact type and \({\mathbb R}\)-rank \(\geq 2\) whenever \(\tilde M_ 1\) is such a space (Appendix 4). The Tits metric is an important new tool in the study of manifolds of nonpositive sectional curvature, and it seems evident that further applications will be found in the future.

In this book Gromov also considers problems of local rigidity; for example, given a simply connected manifold \((\tilde M,g)\) of nonpositive sectional curvature and a compact set \(C\subseteq \tilde M\), is it possible to find a new metric \(g^*\) of nonpositive sectional curvature on \(\tilde M\) that agrees with g on \(\tilde M-C\) and is not isometric to \(g\)? Gromov shows that the answer is negative if \(\tilde M\) satisfies a certain geometric condition which in turn is satisfied by flat Euclidean spaces and by symmetric spaces of noncompact type and \({\mathbb R}\)-rank \(\geq 2\). The basic ideas have recently been extended in V. Schroeder and W. Ziller [Local rigidity of symmetric spaces (preprint, 1986), Trans. Am. Math. Soc. 320, No. 1, 145–160 (1990; Zbl 0724.53033)] to yield similar results for \({\mathbb R}\)-rank 1 symmetric spaces of noncompact type [see also L. Green and R. Gulliver, J. Differ. Geom. 22, 43–47 (1985; Zbl 0563.53036)].

Beyond the scope of this book Gromov has also initiated a study of nonsmooth manifolds of nonpositive curvature that include polyhedra and orbifolds. A manuscript is in preparation.

Reviewer: Patrick Eberlein (Chapel Hill)

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C20 | Global Riemannian geometry, including pinching |

53C35 | Differential geometry of symmetric spaces |